Distribution Theory for the Von Mises-Fisher Distribution and Its Application

Summary

The von Mises-Fisher distribution is the most important distribution in directional data analysis. We derive the sampling distributions of the sample resultant length, the sample mean direction and the component lengths. For the multi-sample case, the conditional distribution of the individual sample resultant lengths given the combined sample resultant length is derived. These results depend heavily on the corresponding distributions for the isotropic random walk on hypersphere. Using these results we investigate some optimum properties of various important tests. Most of these tests were formulated intuitively by Watson and Williams (1956). Mardia (1972) in his book concentrated on the optimum properties of the circular and spherical cases, and this paper extends and unifies some of the parametric work.